Subgroups — #43252624

Order 28544.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383633	C₂	1	2	yes	maximal
2	C₂×#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₂₂₃⋊C₈)	—	2	2	no	—
5	C₄×C₂₂₃⋊C₈	C₄	1	4	yes	—
6	C₂²×C₈×C₂₂₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
8	#97×C₂₂₃	—	2	2	no	—
9	C₂×C₂₂₃⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₂₂₃	—	2	2	no	—
14	C₂×C₈×C₂₂₃	—	2	2	no	—
15	C₂×C₈×C₂₂₃	C₈	1	4	yes	complemented
16	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
17	C₄²×C₂₂₃	D₄	1	2	yes	—
18	C₂₂₃⋊C₈	—	4	4	no	—
19	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	—	4	1	no	—
21	C₂×C₄×C₂₂₃	—	4	1	no	—
22	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
28	C₈×C₂₂₃	—	2	4	no	—
29	C₂²×C₂₂₃	—	2	1	no	—
30	C₂²×C₂₂₃	—	2	1	no	—
31	C₂²×C₂₂₃	#95	1	1	yes	characteristic
32	C₂²×C₂₂₃	C₄⋊C₈	1	2	yes	—
33	C₄×C₂₂₃	—	2	2	no	—
34	C₄×C₂₂₃	—	2	2	no	—
35	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
36	C₄×C₂₂₃	#97	1	1	yes	characteristic
37	C₄×C₂₂₃	C₂²⋊C₈	1	2	yes	—
38	C₂×C₂₂₃	—	2	2	no	—
39	C₂×C₂₂₃	#336	1	1	yes	characteristic
40	C₂×C₂₂₃	H(3,ℤ/4)	1	1	yes	characteristic
41	C₂×C₂₂₃	#338	1	1	yes	characteristic
42	C₂₂₃	#1275	1	1	yes	characteristic, complemented, Sylow
43	#1275	—	223	1	no	maximal, Sylow
44	#431	—	223	2	no	—
45	C₂×#97	D₂₂₃	1	1	yes	characteristic
46	C₂²×C₈	Dic₂₂₃	1	2	yes	—
47	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
48	C₄⋊C₈	—	446	2	no	—
49	C₂²⋊C₈	—	446	2	no	—
50	#97	—	2	2	no	—
51	C₄×C₈	—	223	4	no	—
52	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
53	C₂²×C₄	D₈₉₂	1	1	yes	characteristic
54	C₂²×C₄	Dic₄₄₆	1	1	yes	characteristic
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	446	4	no	—
58	C₂×C₈	C₂₂₃⋊C₈	1	4	yes	complemented
59	C₄²	C₄×D₂₂₃	1	2	yes	—
60	C₄²	C₂²⋊D₂₂₃	1	2	yes	—
61	C₂³	C₄⋊Dic₂₂₃	1	1	yes	characteristic
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
65	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
66	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
67	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
68	C₂×C₄	C₂×C₂₂₃⋊C₈	1	2	yes	—
69	C₂×C₄	#16813154	1	2	yes	—
70	C₈	—	2	4	no	—
71	C₈	—	892	4	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
75	C₂²	C₄⋊(C₂₂₃⋊C₈)	1	2	yes	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
79	C₄	#20297215	1	1	yes	characteristic
80	C₄	C₂²⋊(C₂₂₃⋊C₈)	1	2	yes	—
81	C₂	—	2	2	no	—
82	C₂	#28383335	1	1	yes	characteristic
83	C₂	#28383337	1	1	yes	characteristic
84	C₂	C₂₂₃⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₂₂₃
commutator	C₄×C₂₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	C₂×#97×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	C₂×#97	D₂₂₃
hypocenter	C₂₂₃	#1275
LCS	G ▷ C₄×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
15. C₂×C₈×C₂₂₃	C₈	semidirect
42. C₂₂₃	#1275	semidirect
58. C₂×C₈	C₂₂₃⋊C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#28383633	2
C₂×#97×C₂₂₃	2
#1275	223